Three-dimensional Orr-Sommerfeld equation is analyzed by compound-matrix method to explain the relevance of two- and three-dimensional wall excitations with respect to three-dimensional direct simulation results presented in Sharma and Sengupta, Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation, Phys. Fluids, 31 (6), 064107 (2019). Physical variables are considered in a finite spanwise domain which allows us to consider spanwise wavenumber $\beta$ as the higher harmonics of fundamental spanwise mode ${\beta }_0$ defined by the spanwise domain. Linear stability analysis is performed for various cases with different spanwise wavenumber and neutral curves are generated in both spatial and temporal frameworks. Even though the neutral curves appear different, the critical Reynolds number remains same for all spanwise excitation wavenumbers. The significance of neutral curve in either spatial, or temporal or spatio-temporal frameworks is explained for both modal and non-modal component of the response field. It is shown that as spanwise wavenumber of input excitation increases, the critical Reynolds number increases, while the spatial amplification rate ${\alpha }_i$ decreases. Thus, this establishes that increasing three-dimensionality of input excitation will shown enhanced stability of the flow, and helps one to understand the three-dimensional direct simulation of receptivity analysis for wall excitation reported very recently.

通过复合矩阵方法分析了三维Orr-Sommerfeld方程，以解释二维和三维壁激励与Sharma和Sengupta中提出的三维直接模拟结果，频率和波数对这三个方面的影响的相关性壁激发的三维跃迁路线，物理。流体，31（6），064107（2019）。在有限的展向域中考虑物理变量，这使我们能够考虑展向波数$\beta$ 作为基本展向模式的高次谐波 ${\beta }_0$由spanwise域定义。针对具有不同翼展方向波数的各种情况执行线性稳定性分析，并在空间和时间框架中生成中性曲线。即使中性曲线看起来有所不同，对于所有展向激励波数，临界雷诺数仍保持相同。对于响应场的模态和非模态分量，都说明了中性曲线在空间，时间或时空框架中的重要性。结果表明，随着输入激励的展向波数增加，临界雷诺数增加，而空间放大率增加。${\alpha }_{\mathrm{一世}}$减少。因此，这确定了增加输入激励的三维度将显示出增强的流动稳定性，并有助于人们理解最近报道的壁激励的接受度分析的三维直接模拟。